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Discuss the convergence or divergence of the series \sum _{{n=1}}^{{\infty }}{\frac  {1}{n^{3}}}.

This is a p-series with p=3 so we know that it converges. However, let us see what happens with the ratio test.

\lim _{{n\rightarrow \infty }}\left|{\frac  {{\frac  {1}{(n+1)^{3}}}}{{\frac  {1}{n^{3}}}}}\right|=\lim _{{n\rightarrow \infty }}\left|{\frac  {n^{3}}{n^{3}+3n^{2}+3n+1}}\right|=1\,

With the ratio test giving us a limit of 1, we can make no conclusions by the ratio test. Though we do know this test converges, the ratio test does not give us any information. This is an example that shows us the different tests have their advantages and disadvantages.

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